Quality of Emergence in Video Streaming

• Quality of Emergence (QoE) is a framework that defines end-user experience using perceptual metrics like start-up delay and buffer starvation probabilities.
• It uses continuous-time Markov chains and matrix-based methods to model flow-level dynamics and predict playback interruptions in congested wireless networks.
• The unified framework reveals that flow-level throughput variations are the primary influencers of QoE, while rapid fluctuations from VBR and fast-fading have minimal impact.

Quality of Emergence (QoE) in the context of video streaming is characterized by rigorously defined perceptual metrics that quantify end-user experience under stochastic network and flow dynamics. In wireless networks, playback interruptions—primarily buffer starvation events—emerge from interactions between file transfer and network-level user dynamics. Analytical frameworks have been developed to capture the system-wide and flow-level stochastic processes dictating these quality phenomena, with key metric formulations based on diffusive and Markovian buffer models, and with extensions to account for variable bit rate (VBR) playback and opportunistic scheduling effects. The primary determinants of observed QoE are flow-level throughput variations and state-dependent contention among simultaneous users, whereas rapid fluctuations (e.g., fast-fading or frame-size jitter) are found to have negligible influence on emergent starvation behavior (Xu et al., 2014).

1. Definition and Core Metrics of QoE

Quality of Experience (QoE) for a single streaming session is formalized through two main sets of stochastic metrics:

• Start-up (Prefetching) Delay ($T_a$): The duration from session initiation to playback commencement, with playback only starting after the player buffers $q_a$ seconds of video. Its conditional distribution is

$$\Psi_i(t; q_a) = \Pr\{T_a < t\,|\,I(0) = i\}, \qquad i \in \{0, \ldots, K-1\}$$

where $I(0)$ denotes the number of other flows at admission [Eq. (9)].

• Buffer Starvation Probabilities:
  • Probability of starvation before completion, given buffer state $Q(0)=q$ and initial network state $I(0)=i$:

  $$W_i(q) = \Pr\{\text{starvation before download completes}\,|\,I(0)=i,\,Q(0)=q\}$$

  [Eq. (13)]. - Full probability-generating function (PGF) of the number of starvations is derived (Sec. III-D).

These metrics are computed in the presence of dynamic arrivals ($\lambda$) and departures ($\mu$) of contending users, with admission control threshold $K$ and stationary pre-admission distribution $\pi_i$ [Eq. (8)].

2. Stochastic Flow-Level Modeling and Analytical Formulation

QoE evaluation is founded on a two-tier continuous-time Markov chain (CTMC):

• Birth–Death Dynamics:

  • Before admission, system evolves as a birth–death process for $0, \ldots, K$ flows.
  • On session admission, the system state is indexed by $I(t)\in\{0,\dots,K-1\}$, and evolves with new flow (“birth”) at fixed rate $\lambda_i = \lambda$ (unless $i = K-1$) and departures (“death”) at rate $\nu_i = \frac{i}{i+1}\mu$.
• Prefetching and Playback Rates:
  • The buffer fills at rate $b_i = C/\text{Bitrate} \,(i+1)$ during prefetching ($C$ is capacity), while during playback the buffer “drift” is $c_i = b_i - 1$.

Analytical Structure

• Prefetching Delay Distribution:
  • The buffer draining process is formulated via a linear hyperbolic PDE system:

  $$\frac{\partial U_i}{\partial t} = -b_i \frac{\partial U_i}{\partial q} - (\lambda_i+\nu_i)U_i + \lambda_i U_{i+1} + \nu_i U_{i-1}$$

  [Eq. (19)]. Matrix methods (tridiagonal diagonalization) and Brownian approximations yield solution representations [Eq. (22)].

• Starvation Probabilities:

  • Buffer evolution under playback is captured by ODEs:

  $$c_i \frac{d W_i}{dq} = (\lambda_i + \mu_i) W_i(q) - \lambda_i W_{i+1}(q) - \nu_i W_{i-1}(q)$$

  [Eq. (38)]. The general solution uses eigen-decomposition and exponential integral forms.

• Number of Starvations:

  • The PGF of the total number of starvations is expressed as chained products of matrix exponential solutions for each buffer cycle [Eq. (49)].

3. Extensions to Variable Bit Rate (VBR) and Opportunistic Scheduling

VBR Playback

Playback rate variations on small timescales are modeled as an Itô process:

$d\mathcal{S}(t) = 1 \cdot dt + \bar{\sigma} dB_t$

where $\bar{\sigma}^2$ matches the frame-size variance [Sec. IV-A]. The corresponding buffer-starvation ODE becomes second-order in $q$, incorporating the diffusion coefficient $\frac{\bar{\sigma}^2}{2}$ [Eq. (52)]. The full analytic solution involves $2K\times 2K$ matrix exponentials.

Fast-Fading and Opportunistic Scheduling

When applying opportunistic scheduling (e.g., base station selects user with maximum SNR), the per-slot rate for the tagged flow is a function of the best-user rate $r_i^*$:

$$E[r_i] = \frac{1}{i} E[r_i^*], \qquad \operatorname{Var}[r_i] = \frac{1}{i} \operatorname{Var}[r_i^*] + (E[r_i^*])^2 \left(\frac{1}{i} - \frac{1}{i^2}\right)$$

[Eqs. (64–65)]. By aggregation over scheduling intervals, the mean and variance for the video arrival process are determined. The empirical result is that $\operatorname{Var}[R_i]$ is $\mathcal{O}(10^{-2})$ and thus negligible for starvation prediction; setting $b_i = E[R_{i+1}]$ allows usage of the baseline CBR framework with high accuracy [Figs. 12–13].

4. Analytical Insights and Dominant QoE Determinants

• Dominance of Flow Dynamics:
  • The primary source of QoE degradation—specifically, playback starvation—is due to the flow-level dynamics (user arrivals/departures), which operate on multi-second timescales [Sec. VI].
  • Fast-fading and VBR create only minor, second-order effects owing to their millisecond-scale fluctuations.
• Significance of First Moments:
  • Only the mean throughput and mean playback rate critically determine the buffer starvation distributions and delays. Higher-order moments such as rate variance have limited impact within the examined stochastic process—the corresponding diffusion terms induce only negligible corrections to $W_i(q)$ and other QoE metrics.

A plausible implication is that, for network optimization targeting QoE, mechanisms that regulate number of concurrent flows or modulate resource allocation at the flow level are fundamentally more effective than techniques mitigating rapid physical-layer fluctuations.

5. Numerical Quantification and Empirical Verification

Quantitative findings are presented through simulation and analytical solution comparisons:

• Start-up Delay and Starvation:
  • The starvation probability $P_s(q_a)$ decreases monotonically with increased prefetching threshold $q_a$, and this is consistent over varying congestion levels and bitrates [Fig. 6].
  • Starvation probability is near one in highly congested states at playback initiation, except for extremely large $q_a$ [Fig. 7].
  • The PMF of the number of starvations aligns closely between matrix-product PGF calculations and simulation [Fig. 8].
• Solution Accuracy:
  • The numerical PDE solution yields the most accurate CDF of start-up delays, with Brownian/erfc approximations retaining fidelity for mean delay [Fig. 9].
• VBR and Fast-Fading Results:
  • Even with frame-size variance $\bar{\sigma}^2$ up to 0.5, starvation probabilities deviate negligibly from the CBR case [Figs. 10–11].
  • Under Rayleigh fast-fading with normalized SNR scheduling, the basic CBR model predicted starvation probability to within less than 1% error across states [Figs. 12–13].

6. Unified Framework and Methodological Summary

The unified stochastic framework comprises:

• PDE-based methods for prefetching delay estimation.
• ODE-based methods for buffer starvation probabilities.
• Matrix-exponential and probability-generating functions for computing starvations and transitions.

These approaches together capture (i) random user join/leave events, (ii) closed-form evaluation of key QoE statistics under constant bit rate streaming, and (iii) natural extensibility to VBR playback and fast-fading wireless channels by modification of the mean rates and inclusion of diffusive corrections. Empirical and analytic evidence supports that flow-level variability is the fundamental limiting factor for video streaming QoE in wireless network scenarios (Xu et al., 2014).